Define A a0 a2x2 a2n x2n the family of polynomials cont

Define A_+ = {a_0 + a_2x^2 + + a_2n x^2n} - the family of polynomials containing only even powers of x. Similarly, A_- = {a_1 x + a_3x^3 + + a_2n+1 X^2n + 1} is the collection of polynomials containing only odd powers of x. Is A_+ dense in C[0, 1]? Is A_- dense in C[-1, 1]? Is {p(x)^2} - the collection of squares of polynomials - dense in C[0, 1]?

Solution

(1).

The every continuous function defined on a closed interval [0, 1] can be uniformly approximated as closely as desired by a polynomial function.

The polynomial functions (even powers of x )are dense in [0,1], and each polynomial function can be uniformly Approximated by one with rational coefficients.

(2)

The every continuous function defined on a closed interval [-1, 1] can be uniformly approximated as closely as desired by a polynomial function.

The polynomial functions (odd powers of x )are dense in [-1, 1], and each polynomial function can be uniformly Approximated by one with rational coefficients.